ICSV14 Cairns • Australia

9-12 July, 2007

AN ASSESSMENT OF ALTERNATIVE SENSOR TECHNOLOGYFOR TRANSFER MATRIX MEASUREMENTS IN COMBUSTORS

Jonas P. Moeck, Holger J. Konle, Christian Oliver Paschereit

Institut für Strömungsmechanik und Technische AkustikTechnische Universität Berlin

10623 Berlin, Germanyjonas.moeck@pi.tu-berlin.de

Abstract

Acoustic transfer matrices are an important modeling tool for muffler design and thermoacous-tic system analysis. They allow for a relatively simple and modular description of (thermo-)acoustic systems in which only plane waves propagate. However, in case of combustion sys-tems, where the flame transfer matrix (or the scalar flame transfer function) is one of the mostimportant modeling inputs, measuring conditions are quite harsh due to high temperatures (wellabove 1000 ◦C) and possibly high static pressures. The applicability of two high-temperature re-sistant pressure sensors to transfer matrix measurements is, therefore, assessed in this work. Thefirst is a probe microphone designed according to the semi-infinite coil principle. As a secondalternative, a fiber optic microphone is considered. This less common pressure sensor is builtfrom a thin metallic membrane and a vibrometer detecting the transversal displacement. Twoacoustic elements are considered, a uniform duct and a circular orifice. The transfer matrices ob-tained using the two alternative sensors are compared to measurements with wall flush-mountedstandard 1/4" condenser microphones. The results show reasonable agreement and demonstratethe general applicability of the alternative sensors to transfer matrix measurements.

1. INTRODUCTION

Acoustic transfer matrices are a useful concept in various areas of application. Modeling of muf-flers [1] and thermoacoustic systems [2] are only two examples were extensive use is made ofthis approach. The advantage lies in a very simple acoustic representation of an element where,at the inlet and at the outlet, only plane waves propagate. In addition to that, it is relatively easyto connect several elements and obtain the acoustic response for a combined system or modela lumped boundary condition of a more complex geometrical set up. Even flow-acoustic andflow-combustion-acoustic interactions in the linear regime can be modeled using this approach(see, e.g., Hofmans et al. [3] and Schuermans et al. [4]). However, if transfer matrices containinformation about elements with flow or combustion, analytical or numerical access might bevery limited. In these cases, experimental methods have to be used.

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In the last two decades, the so called Multi-Microphone-Technique (MMM) [4–6] hasbeen proven to be the state of the art for this task. The usual experimental procedure is to useseveral condenser microphones (either 1/4" or 1/2") to assess the acoustic pressure at severalaxial locations. This is necessary to find the four scalar complex valued functions of frequencythat constitute the transfer matrix. In case of combustion systems, for which the flame trans-fer matrix (or the scalar flame transfer function) is one of the most important modeling inputs[7, 8], measuring conditions are quite harsh due to very high temperatures (up to 1600 ◦C) andpossibly high static pressures. Flame transfer matrix measurements are therefore accomplishedusing water-cooled microphone holders [4, 5, 7]. These are difficult to design with no impacton the pressure frequency response and still cannot fully protect the microphone against harshcombustor conditions. Also, a condenser microphone is very sensible to electromagnetic inter-ference.

For these reasons, two alternative methods to assess the pressure under harsh conditionsfor transfer matrix characterization are investigated here. The first makes use of a probe holderbuilt upon the semi-infinite coil principle. In this approach, the actual pressure sensor is re-motely located outside the combustor, and the pressure is guided through a thin tube. Suchdevices were built by several researchers for dynamic pressure measurements in combustors[9–11]. To our knowledge, however, none of them tried to apply this type of sensor to trans-fer matrix measurements. One potential problem is that the frequency response of these probemicrophones with respect to the wall pressure is always affected by the remoting system. Thesecond sensor is a fiber optic microphone, a combination of a laser vibrometer and a high tem-perature resistant reflecting diaphragm.

2. HIGH-TEMPERATURE RESISTANT PRESSURE SENSORS

2.1. Fiber optic microphone

Laser

Interferometer

Cooling air

Fiber

Acoustic fieldMembrane

Figure 1. Fiber optic microphone princi-pal set-up

The fiber optic microphone (FOM) used in this study isstill under development, but to evaluate its current sta-tus with respect to applications, transfer matrix mea-surements were conducted. A reflecting membrane, aglass fiber, and an interferometer are the main com-ponents of the FOM. The vibrometer (glass fiber andinterferometer) detects surface displacement by meansof laser beam interference. The spatial separation ofmembrane and transducer electronics makes this sensoruniquely adequate for the application to pressure mea-surements in high temperature or even EMI (ElectroMagnetic Interference) or RFI (Radio Frequency Interference) environments. In the frequencyrange considered here, the FOM exhibits a noisy but approximately flat frequency response withrespect to a wall flush-mounted condenser microphone (see Fig. 2). Also, it can be noted, thatthe FOM delivers the unsteady pressure with no delay, which makes it particularly attractive asa sensor for control applications. More details on the FOM, including the dependence of theacoustic response on design parameters as, e.g., border tension, as well as the linearity of theresponse were presented by Konle et al. [12].

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0.5

1

1.5

2

200 400 600 800 1200

0

frequency frequencyHzHz

|F|

arg

F

-π

-π

2

-π

-π

2

Figure 2. FOM transfer function with respect to wall flush-mounted microphone

2.2. Probe microphone

coil len

gth

Lb

La

by courtesy of DLR Berlin

Figure 3. Probe microphone design

The probe holder considered in this work was designedand manufactured by the German Aerospace Center, In-stitute of Propulsion Technology, Berlin, and has beenused in a number of combustion instability characteriza-tion and control investigations (see, e.g., Bake et al. [13]or Moeck et al. [14]). Figure 3 displays the design of theprobe holder. The microphone is acoustically connected tothe measurement location via a steel tube (2 mm diameter).To avoid resonances in the connecting duct, the microphoneis mounted perdendicularly to the tube, and the tube is elon-gated to approximately 1 m. In this way, visco-thermal ef-fects significantly attenuate the acoustic waves and, thereby,minimize reflections from the tube end. Due to this set up,the response of the microphone in the probe assembly willbe different compared to a measurement with a wall flush-mounted microphone. The trans-fer function of the probe microphone with respect to a flush-mounted sensor is displayed inFig. 4. The results for three different probe holders are presented, showing virtually no differ-ence in the response up to 1 kHz. In addition to the experimental data, the transfer function ofa transmission line model of the probe microphone is also added. A straightforward plane wavecalculation shows that the transfer function of the probe holder can be written as

F =p1

p0

=eLakΓ

(e2LbkΓ + R

)e2(La+Lb)kΓ + R

, (1)

where k denotes the wave number of the plane mode, La is the distance between measurementlocation and microphone, Lb is the coil length, R the reflection coefficient at the end of the coil(which has been assumed to be −1), and p0 and p1 denote wall pressure and pressure in thetransmission line at the microphone position, respectively. Γ represents the complex (frequencydependent) propagation constant taking into account visco-thermal damping. Here, Γ has beencalculated from the low reduced frequency approximation of the full Kirchhoff solution (see,e.g., Tijdeman [15]). The oscillatory behavior of the transfer function (Fig. 4) is due to wavereflections at the end of the coil, which are not damped completely. As a result of the spatial

ICSV14 • 9–12 July 2007 • Cairns • Australia

0 200 400 600 10000

0.5

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1.5

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0 200 400 600 1000

0

frequency frequencyHzHz

|F|

arg

F

-π

-π

2

-π

-π

2

− probe 1 − probe 3− probe 2 − Eq. (1)

Figure 4. Probe microphone transfer function; model and experimental data

separation of measurement location and microphone position, there is a time delay of La/c (cdenoting speed of sound) in the probe response with respect to a wall flush-mounted micro-phone, corresponding to the negative slope of the linear decrease in the phase response (Fig. 4).There is good agreement of the modeled transfer function of the probe holder (Eq. (1)) andthe experimental data. Detailed information on the probe microphone was presented by Forsteret al. [16], as, e.g., the influence of a purge flow in the coil on the frequency response.

3. TRANSFER MATRIX MEASUREMENTS

The acoustic transfer matrix is defined as a 2×2 mapping of acoustic variables representingthe plane wave mode. This mapping provides frequency response information about a certainacoustic element. Strictly speaking, the transfer matrix maps acoustic pressure p and axial parti-cle velocity u at the upstream location of the element to pressure and velocity at the downstreamlocation. This can be written as [

pd

ud

]=

[T11 T12

T21 T22

] [pu

uu

], (2)

where subscripts u and d denote up- and downstream locations, respectively. Here, as in thefollowing, the acoustic pressure is assumed to be scaled by the characteristic impedance ρc, ρ

being the fluid density. The elements of the transfer matrix are complex valued functions offrequency. An equivalent description is given by the scattering matrix, which defines a map-ping between the Riemann invariants. The Riemann invariants are, simply speaking, the waveamplitudes of the up- and downstream traveling waves. The relation to the primitive acousticvariables is given by

p

ρc= f + g, u = f − g, (3)

where f and g represent down- and upstream traveling waves, respectively. Accordingly, thescattering matrix S maps the incident waves to the outgoing, viz.,[

fd

gu

]=

[S11 S12

S21 S22

] [fu

gd

]. (4)

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To determine the four transfer (or scattering) matrix elements, at least two independentacoustic states are required. One way to generate the two states is to excite the acoustic fieldat up- and downstream locations. Acoustic pressure and velocity or, equivalently, the Riemanninvariants can be determined from pressure measurements at multiple (at least two) axial loca-tions. This is called the Multi-Microphone-Method (MMM) [4, 17]. Using only microphonesto determine the plane wave acoustic field has the advantage that no velocity probe is neces-sary. Measuring the acoustic particle velocity (usually associated with very small amplitudes)directly is difficult and, therefore, error prone.

In a duct of constant cross-section, the frequency domain plane wave acoustic field in aquiescent fluid can be written as [1]

p(x, ω) = f(ω) exp (−ikx) + g(ω) exp (ikx), (5a)

u(x, ω) = f(ω) exp (−ikx)− g(ω) exp (ikx), (5b)

where k = ω/c is the wave number of the plane wave field (neglecting visco-thermal damping)and i =

√−1. Given the experimentally determined pressure phasors at multiple axial locations,

Eq. (5) can be formally inverted in a least squares sense to yield

[f(ω)

g(ω)

]= Z+

p(x1, ω)

p(x2, ω)...

p(xn, ω)

, with Z =

exp (−ikx1) exp (ikx1)

exp (−ikx2) exp (ikx2)...

...exp (−ikxn) exp (ikxn)

, (6)

and (·)+ denotes the pseudoinverse. Once f and g are known, the plane wave pressure field in aduct of constant cross-section is completely determined, and p and u at an arbitrary location (inthat duct) can be calculated.

Given now complex wave amplitudes up- and downstream of the element for two differentacoustic states, the scattering matrix can be calculated from

S =

[fA

d fBd

gAu gB

u

] [fA

u fBu

gAd gB

d

]−1

, (7)

where superscripts A and B correspond to the two independent excitation states. In view ofEq. (3), the transfer matrix can be computed in a similar way.

3.1. Transfer matrix measurements using wall flush-mounted condenser microphones

In standard transfer matrix measurements for muffler or flame characterization in the linearregime, several 1/4" microphones are used simultaneously to assess the acoustic pressure atdifferent axial positions, flush-mounted to the channel wall. To determine frequency domainphasors, the data is usually correlated with the excitation signal and averaged to suppress ran-dom noise contributions [6]. A proper calibration for the microphones is necessary to identifythe plane wave field. This can be either a one-point-calibration using a pistonphone at one fre-quency or a relative calibration, where all microphones are exposed to the same acoustic fieldand all Fourier spectra are referenced to one baseline microphone. In the latter case, frequencydependent calibration coefficients are obtained, that include magnitude and phase information.

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The latter method is more accurate, in particular in the low frequency regime, and allows toreliably detect faulty sensors.

Two typical excitation and signal processing techniques are frequency sweeps and steppedsine in combination with cross-spectrum based frequency domain averaging. However, variousalternative possibilities exist (see, e.g., Allam & Bodén [6] and references therein). Since theexperiments where conducted in an environment with a low noise level, sweep excitation wasused. Note, however, that in combustors with high thermal power, stepped sine excitation is theonly way to obtain reasonable results.

3.2. Transfer matrix measurements using FOM and probe microphone

Since only one FOM prototype was available, the acoustic pressure for one excitation state couldnot be acquired simultaneously at different axial positions. For this reason, the same FOM wasmounted to the channel at three different locations successively, on the up- and downstreamside, and was exposed to the same acoustic field at each position, corresponding to the twoindependent excitation states. To allow for a meaningful comparison, the same procedure wasused for the probe microphone. The axial locations for the sensor mountings were the sameas in the case of measurements using condenser microphones. Using only one sensor for allmeasurement locations, it was not necessary to take into account the frequency response ofthe FOM and the probe microphone (i.e., a calibration) with respect to a wall flush-mountedcondenser microphone since this cancels out in the transfer matrix computation.

4. EXPERIMENTAL SET-UP

The experiments were conducted in the acoustic test facility shown in Fig. 5. Up to 80 micro-phones in total can be mounted at ten different axial and eight different azimuthal positions.However, since in this study plane wave properties were studied, only one azimuthal positionwas used. Also, only three axial positions each, up- and downstream of the acoustic element,were equipped with microphones. The axial measurement positions and the location of theacoustic element are shown in Fig. 5. The duct is made of aluminum with a wall thickness of10 mm and has a diameter of 140 mm. Accordingly, the cut-on frequency for the first non-planarmode is at f10 = 1435 Hz at ambient conditions [1]. The up- and downstream ends of the testrig are equipped with anechoic terminations.

Figure 5. Experimental set up; speakers for acoustic excitation (marked blue), axial measurement posi-tions (marked black) and acoustic element for transfer/scattering matrix determination (marked red)

ICSV14 • 9–12 July 2007 • Cairns • Australia

5. RESULTS AND DISCUSSION

Two generic acoustic elements were considered in this study, a uniform duct of length 400 mmand a circular orifice with inner diameter 40 mm and thickness 3 mm. In case of the uniformduct, the scattering matrix is considered since it has a remarkably easy representation. Theacoustic waves travel through the element without reflection. Only the phase is changed due tothe propagation time, which is given by τ = L/c, where L is the duct length. Therefore, thescattering matrix for a uniform duct simply reads

Sduct =

[e−iωτ 0

0 e−iωτ

]. (8)

Due to lack of space, not the full scattering and transfer matrices will be presented butonly one representative element. Figure 6 shows the measured upper left scattering matrix ele-ment for the uniform duct (left column). There is perfect agreement with the theoretical result(Eq. (8)) in case of the wall flush-mounted microphones (top left). The only deviation that canbe observed appears close to 1400 Hz. This can be attributed to higher, non-planar modes, whichare excited by the laterally mounted speakers and decay at a slow rate in axial direction close tothe cut-on frequency.

The S11 element as obtained from measurements using the FOM is shown in the middleleft frame of Fig. 6. The results are markedly not as good as in the case of condenser microphonemeasurements. The magnitude is about 10 % too small, but the slope of the phase is accuratelycaptured. Also, the data scatter is slightly higher. Altogether, there is still reasonable agreement

0

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|S11|

|S11|

|S11|

arg

S11

arg

S11

arg

S11

<(T

12)

<(T

12)

<(T

12)

=(T

12)

=(T

12)

=(T

12)

HzHz

-π

-π

-π

-π

-π

-π

1a©

1b©

1c©

2a©

2b©

2c©

Figure 6. Left column: S11-element of the duct scattering matrix. Right column: T12-element of theorifice transfer matrix. Top row: wall flush-mounted microphones. Middle row: FOM. Bottom row: probemicrophone. Solid lines correspond to the theoretical result (Eq. (8)) in case of the duct and to the finiteelement calculation in case of the orifice. Symbols represent measured data

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with the theoretical solution. The results from the probe microphone are more accurate (bottomleft in Fig. 6). Magnitude and phase of S11 are properly represented. A higher data scatter com-pared to the measurements with wall flush-mounted microphones (Fig. 6, top left) is apparent,though, in particular between 800 and 1000 Hz.

The results for the T12-element of the transfer matrix of the circular orifice are presentedin the right column of Fig. 6. For comparison, T12 as obtained from a finite element computationbased on the Helmholtz equation (see, e.g., Peat [18]) is also shown. Note that this is in perfectagreement with the so-called L − ζ model [5] for an acoustically compact element, which,in case of equal cross-sections up- and downstream of the element and with no mean flow, isgiven by T12 = −ikLeff. Again, the results from the condenser microphone measurements agreequite well with those from the finite element solution. The transfer matrix element obtainedwith the FOM is significantly more scattered. The general trend of real and imaginary partscan be recognized but the results are certainly not sufficiently accurate to apply this sensor tothe determination of unknown transfer matrices. In contrast, the T12 element of the orifice iswell obtained from the measurements with the probe microphone. The results are not as goodas those from the wall flush-mounted pressure sensors, but the real and imaginary part clearlyagree with the FEM solution. In case of the FOM and the probe microphone measurements, acoherence filter was applied to the measured scattering and transfer matrices. Only the resultsat those frequencies were plotted, where the coherence of all sensor signals (at the six axiallocations) with respect to the excitation command was close to unity.

The crucial point in the determination of the scattering (or transfer) matrix is the identi-fication of the f and g waves from the complex pressure amplitudes at the axial measurementlocations. As a quality indicator of this identification process, the normalized wave identifica-tion residual (WIR) δ is defined as

δ =|| (I − ZZ+) p||

||p||, (9)

where Z is given in Eq. (6) and p is a vector containing the complex pressure amplitudes cor-responding to one excitation state and one microphone group (either up- or downstream of theelement). I is the identity matrix of dimension n (in this case n = 3) and || · || denotes thel2-norm. For n > 2 and incommensurate microphone spacings, δ can only vanish if there existcomplex values of f and g, such that Eq. (5a) is satisfied for all complex pressure amplitudesin one group. δ maps to values in [0, 1], where low values indicate a good wave identificationprocess.

The WIR for the duct scattering matrix measurements with condenser microphones, FOM,and probe microphone is displayed in Fig. 7. The four curves in one plot correspond to the twoexcitation states, each with two microphone groups, up- and downstream of the element, respec-tively. For the condenser microphone measurements, the WIR has values of a few percent up to1200 Hz, indicating an accurate wave identification process. Larger values of δ for frequenciesgreater than 1200 Hz can again be attributed to the first circumferential mode. The WIR for theFOM measurements has much larger values over the whole frequency range considered. Thismeans that the wave identification process has a much lower quality compared to the condensermicrophone measurements and is consistent with the data scatter in Fig. 6 (middle row). Incase of the probe microphone measurements, the WIR is significantly lower, indicating a moreaccurate identification of the f and g waves from the measured pressure phasors. The highest

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200 400 600 800 1000 140010

−3

10−2

10−1

100

200 400 600 800 1000 1400 200 400 600 800 1000 140010

−3

10−2

10−1

100

frequencyfrequencyfrequency

δ δ

HzHzHz

microphones FOM probe microphone

Figure 7. Wave identification residual for measurements with condenser microphones (left), FOM (mid-dle) and probe microphone (right). Different lines correspond to the microphone groups for the twoexcitation states, up- and downstream of the element, respectively

values are attained at frequencies between 800 and 1000 Hz (disregarding the increase closeto the cut-on frequency for the first circumferential mode). This coincides with the frequencyrange where the data scatter in S11 is largest (see left middle frame in Fig. 6). Also, it can benoted that the probe microphone transfer function (Fig. 4) is somewhat contaminated with noisein this frequency regime.

As stated in Sec. 3.2, when using only one sensor at different axial measurement locations,the transfer function with respect to the wall pressure cancels out when computing the scatteringor transfer matrix. Therefore, this cannot be a source of the high data scatter in the resultsfrom the FOM measurements. Errors might have been introduced through the mounting anddemounting of the sensor at the different axial locations. On the other hand, the same procedurewas used for the probe microphone, where clearly more accurate results were obtained. A sourceof errors could be a not fully linear response of the FOM (see Konle et al. [12]).

6. CONCLUSIONS

The principal applicability of two high-temperature resistant pressure sensors, a fiber optic mi-crophone and a probe microphone, to transfer matrix measurements was studied. Two genericacoustic elements were considered, a uniform hard-walled duct and a circular orifice. Reason-able results were obtained with both sensors. In case of the probe microphone, the measuredtransfer matrices were only slightly less accurate than those obtained from conventional wallflush-mounted pressure sensors. Although the fiber optic microphone showed satisfactory re-sults for the straight duct, high data scatter contaminated the transfer matrix in case of theorifice.

The results presented indicate that the probe microphone can be used to measure transfermatrices in high-temperature environments. However, in the presence of flow and combustion,there is a much higher noise level than considered in this study. The fiber optic microphoneclearly needs further improvement before it can be utilized for accurate measurements in thisapplication.

ACKNOWLEDGEMENTS

We kindly thank the combustion group of the German Aerospace Center, Berlin, for makingthe probe microphones available to us. Financial support from the German Science Foundationthrough the Collaborative Research Center 557 “Control of Complex Turbulent Shear Flows” as

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well as from the Helmholtz Association as part of the Young Investigators Group “New OpticalMeasurement Techniques for Turbomachinery Diagnostics” is gratefully acknowledged.

REFERENCES

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[5] SCHUERMANS, B., POLIFKE, W., AND PASCHEREIT, C. O., 1999. “Modeling of transfer matrices ofpremixed flames and comparison with experimental results”. ASME Paper 1999-GT-0132.

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[14] MOECK, J. P., BOTHIEN, M. R., GUYOT, D., AND PASCHEREIT, C. O., 2006. “Phase-shift control ofcombustion instability using (combined) secondary fuel injection and acoustic forcing”. In Proc. 1st Conf.Active Flow Control, Berlin, Germany.

[15] TIJDEMAN, H., 1975. “On the propagation of sound waves in cylindrical tubes”. J. Sound Vib., 39(1),pp. 1–33.

[16] FORSTER, S., RÖHLE, I., AND MICHEL, U., 2004. Optimierung der passiven und aktiven Dämpfung vonthermoakustischen Schwingungen. Final report AG Turbo II project 4.4.2.C, Deutsches Zentrum für Luft-und Raumfahrt, Berlin.

[17] PASCHEREIT, C. O., SCHUERMANS, B., POLIFKE, W., AND MATTSON, O., 2002. “Measurement of trans-fer matrices and source terms of premixed flames”. J. Eng. Gas Turb. Power, 124, pp. 239–247.

[18] PEAT, K. S., 1982. “Evaluation of four-pole parameters for ducts with flow by the finite element method”.J. Sound Vib., 84(3), pp. 389–395.